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GROUPOID SCHEMES 022L Contents 1. Introduction 1 2. Notation 1 3. Equivalence relations 2 4. Group schemes 4 5. Examples of group schemes 5 6. Properties of group schemes 7 7. Properties of group schemes over a field 8 8. Properties of algebraic group schemes 14 9. Abelian varieties 18 10. Actions of group schemes 21 11. Principal homogeneous spaces 22 12. Equivariant quasi-coherent sheaves 23 13. Groupoids 25 14. Quasi-coherent sheaves on groupoids 27 15. Colimits of quasi-coherent modules 29 16. Groupoids and group schemes 34 17. The stabilizer group scheme 34 18. Restricting groupoids 35 19. Invariant subschemes 36 20. Quotient sheaves 38 21. Descent in terms of groupoids 41 22. Separation conditions 42 23. Finite flat groupoids, affine case 43 24. Finite flat groupoids 50 25. Descending quasi-projective schemes 51 26. Other chapters 52 References 54 1. Introduction 022M This chapter is devoted to generalities concerning groupoid schemes. See for exam- ple the beautiful paper [KM97] by Keel and Mori. 2. Notation 022N Let S be a scheme. If U, T are schemes over S we denote U(T ) for the set of T -valued points of U over S. In a formula: U(T ) = MorS(T,U). We try to reserve This is a chapter of the Stacks Project, version fac02ecd, compiled on Sep 14, 2021. 1 GROUPOID SCHEMES 2 the letter T to denote a “test scheme” over S, as in the discussion that follows. Suppose we are given schemes X, Y over S and a morphism of schemes f : X → Y over S. For any scheme T over S we get an induced map of sets f : X(T ) −→ Y (T ) which as indicated we denote by f also. In fact this construction is functorial in the scheme T/S. Yoneda’s Lemma, see Categories, Lemma 3.5, says that f determines and is determined by this transformation of functors f : hX → hY . More generally, we use the same notation for maps between fibre products. For example, if X, Y , Z are schemes over S, and if m : X ×S Y → Z ×S Z is a morphism of schemes over S, then we think of m as corresponding to a collection of maps between T -valued points X(T ) × Y (T ) −→ Z(T ) × Z(T ). And so on and so forth. We continue our convention to label projection maps starting with index 0, so we have pr0 : X ×S Y → X and pr1 : X ×S Y → Y . 3. Equivalence relations 022O Recall that a relation R on a set A is just a subset of R ⊂ A × A. We usually write aRb to indicate (a, b) ∈ R. We say the relation is transitive if aRb, bRc ⇒ aRc. We say the relation is reflexive if aRa for all a ∈ A. We say the relation is symmetric if aRb ⇒ bRa. A relation is called an equivalence relation if it is transitive, reflexive and symmetric. In the setting of schemes we are going to relax the notion of a relation a little bit and just require R → A × A to be a map. Here is the definition. 022PDefinition 3.1. Let S be a scheme. Let U be a scheme over S. (1) A pre-relation on U over S is any morphism of schemes j : R → U ×S U. In this case we set t = pr0 ◦ j and s = pr1 ◦ j, so that j = (t, s). (2) A relation on U over S is a monomorphism of schemes j : R → U ×S U. (3) A pre-equivalence relation is a pre-relation j : R → U ×S U such that the image of j : R(T ) → U(T ) × U(T ) is an equivalence relation for all T/S. (4) We say a morphism R → U ×S U of schemes is an equivalence relation on U over S if and only if for every scheme T over S the T -valued points of R define an equivalence relation on the set of T -valued points of U. In other words, an equivalence relation is a pre-equivalence relation such that j is a relation. 02V8 Lemma 3.2. Let S be a scheme. Let U be a scheme over S. Let j : R → U ×S U be a pre-relation. Let g : U 0 → U be a morphism of schemes. Finally, set 0 0 0 0 j 0 0 R = (U ×S U ) ×U×S U R −→ U ×S U Then j0 is a pre-relation on U 0 over S. If j is a relation, then j0 is a relation. If j is a pre-equivalence relation, then j0 is a pre-equivalence relation. If j is an equivalence relation, then j0 is an equivalence relation. Proof. Omitted. GROUPOID SCHEMES 3 02V9Definition 3.3. Let S be a scheme. Let U be a scheme over S. Let j : R → U ×S U be a pre-relation. Let g : U 0 → U be a morphism of schemes. The pre-relation 0 0 0 0 0 j : R → U ×S U is called the restriction, or pullback of the pre-relation j to U . 0 In this situation we sometimes write R = R|U 0 . 022Q Lemma 3.4. Let j : R → U ×S U be a pre-relation. Consider the relation on points of the scheme U defined by the rule x ∼ y ⇔ ∃ r ∈ R : t(r) = x, s(r) = y. If j is a pre-equivalence relation then this is an equivalence relation. Proof. Suppose that x ∼ y and y ∼ z. Pick r ∈ R with t(r) = x, s(r) = y and pick r0 ∈ R with t(r0) = y, s(r0) = z. Pick a field K fitting into the following commutative diagram κ(r) / K O O κ(y) / κ(r0) Denote xK , yK , zK : Spec(K) → U the morphisms Spec(K) → Spec(κ(r)) → Spec(κ(x)) → U Spec(K) → Spec(κ(r)) → Spec(κ(y)) → U Spec(K) → Spec(κ(r0)) → Spec(κ(z)) → U By construction (xK , yK ) ∈ j(R(K)) and (yK , zK ) ∈ j(R(K)). Since j is a pre- equivalence relation we see that also (xK , zK ) ∈ j(R(K)). This clearly implies that x ∼ z. The proof that ∼ is reflexive and symmetric is omitted. 0DT7 Lemma 3.5. Let j : R → U ×S U be a pre-relation. Assume (1) s, t are unramified, (2) for any algebraically closed field k over S the map R(k) → U(k) × U(k) is an equivalence relation, (3) there are morphisms e : U → R, i : R → R, c : R ×s,U,t R → R such that U / R R / R R ×s,U,t R / R e i c ∆ j j j j×j j flip pr02 U ×S U / U ×S UU ×S U / U ×S UU ×S U ×S U / U ×S U are commutative. Then j is an equivalence relation. Proof. By condition (1) and Morphisms, Lemma 35.16 we see that j is a unram- ified. Then ∆j : R → R ×U×S U R is an open immersion by Morphisms, Lemma 35.13. However, then condition (2) says ∆j is bijective on k-valued points, hence ∆j is an isomorphism, hence j is a monomorphism. Then it easily follows from the commutative diagrams that R(T ) ⊂ U(T ) × U(T ) is an equivalence relation for all schemes T over S. GROUPOID SCHEMES 4 4. Group schemes 022R Let us recall that a group is a pair (G, m) where G is a set, and m : G × G → G is a map of sets with the following properties: (1) (associativity) m(g, m(g0, g00)) = m(m(g, g0), g00) for all g, g0, g00 ∈ G, (2) (identity) there exists a unique element e ∈ G (called the identity, unit, or 1 of G) such that m(g, e) = m(e, g) = g for all g ∈ G, and (3) (inverse) for all g ∈ G there exists a i(g) ∈ G such that m(g, i(g)) = m(i(g), g) = e, where e is the identity. Thus we obtain a map e : {∗} → G and a map i : G → G so that the quadruple (G, m, e, i) satisfies the axioms listed above. A homomorphism of groups ψ :(G, m) → (G0, m0) is a map of sets ψ : G → G0 such that m0(ψ(g), ψ(g0)) = ψ(m(g, g0)). This automatically insures that ψ(e) = e0 and i0(ψ(g)) = ψ(i(g)). (Obvious notation.) We will use this below. 022SDefinition 4.1. Let S be a scheme. (1) A group scheme over S is a pair (G, m), where G is a scheme over S and m : G ×S G → G is a morphism of schemes over S with the following property: For every scheme T over S the pair (G(T ), m) is a group. (2) A morphism ψ :(G, m) → (G0, m0) of group schemes over S is a morphism ψ : G → G0 of schemes over S such that for every T/S the induced map ψ : G(T ) → G0(T ) is a homomorphism of groups. Let (G, m) be a group scheme over the scheme S. By the discussion above (and the discussion in Section 2) we obtain morphisms of schemes over S: (identity) e : S → G and (inverse) i : G → G such that for every T the quadruple (G(T ), m, e, i) satisfies the axioms of a group listed above. Let (G, m), (G0, m0) be group schemes over S. Let f : G → G0 be a morphism of schemes over S.
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